14:00h – 14:40h

A result of multiplicity of nontrivial solutions to an elliptic Kirchhoff-Boussinesq equation type in $\mathbb{R}^{N}$

In this work, using  Ljusternik-Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions  for the following class of elliptic Kirchhoff-Boussinesq type problems given by

$$\varepsilon^4 \Delta^2 u \pm \varepsilon^p \Delta_p u + V(x)u=f(u) + \beta|u|^{2_{**}-2}u ~~\text{in}~~\mathbb{R}^N,\\ u \in H^2(\mathbb{R}^N),$$

where $\varepsilon>0$, $2<p<2^{*}=\frac{2N}{N-2}$,  $2_{**}=\frac{2N}{N-4}$ and $N\geq5$, $V:\mathbb{R}^N \to\mathbb{R}$ is a continuous function and $f:\mathbb{R} \to \mathbb{R}$ is a function of $C^{1}$ class. We consider the subcritical case, i.e, $\beta=0$ and critical case, i.e, $\beta=1$.

14:50h – 15:30h

Roughness-induced effects on a reaction-diffusion equation defined in a thin domain

 In this talk we discuss a reaction-diffusion problem in a thin plane region, endowed with a Robin-type boundary condition, which describes reactions catalyzed at the boundary. Motivated by microfluidic applications we allow, in principle, resonant rough behavior in which the amplitude and period of the roughness at the boundary have the same scale as the domain thickness.